Research On Procrustes Problems For Two-sided Orthogonal And P-exchange Of Natrix Equations

Procrustes problems is one of important issues in the field of numerical alge-bra.Actually, it is widely used in many fields such as control theory, dynamic program-ming, statistics, transportation theory, and engineering computation field. We denoterespectively OR^m×n,SR^m×n and SOR^n×n as m×n orthogonal matrix , m×n symmetric matrix and n×nsymmetric orthogonal matrix, where·stands for the Frobenius norm.This thesis is mainly concerned with the problems about two kinds of procrustesproblems.ProblemⅠGiven two matrices A∈R^m×m, B∈R^m×m , find X∈OR^m×m suchthat||X^TAX - B|| = min.ProblemⅡGiven two matrices A∈R^m×m, B∈R^n×n , find X∈OR^m×n,m > n, such that||X^TAX - B|| = min.ProblemⅢGiven two matrices A, B∈R^m×n, P∈SR^n×n, find X∈SP（1）such that||AX - B ||= min,where SP（1 ）= {X|XP = PX, X∈OR^n×n}.ProblemⅣGiven two matrices A, B∈R^m×n, P∈SR^n×n, find X∈SP（2）such that||AX - B|| = min,where SP（2 ）= {X|XP = PX, X∈SOR^n×n}.We call problem I and problemⅡfor two-sided orthogonal Procrustes problems,problemⅢand problem IV for P-exchange Procrustes problems.This paper mainly research works are as follows:1. About two-sided orthogonal procrustes problems, on the feasible set , As theprojected gradient matrices can naturally be formulated by matrices, we presents a wayto the projected Hessian.Using this calculation, we get respectively the complete char-acterization of the first order and the second order optimality condition for Two-sidedorthogonal procrustes problem. In view of the projected gradient for formulation, it can serve as a globally convergent numerical method.The proposed approach isillustrated by numerical examples.2. About P-exchange procrustes problems, general solution of P-exchange orthog-onal procrustes problem and P-exchange symmetrical orthogonal procrustes problemcan naturally be solved by the singular value decomposition of matrix and the matrix ofthe nature of trace. While set of Solution is not empty , we also obtained the nearness.Finally, we illustrated the main theories and algorithm with numerical examples.